#8471779

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,8)(4,7)(5,6)(9,10,12,14)(11,13,15,16)(20,28)(24,29)(26,32)(30,31);(2,6)(3,7)(11,15)(13,16)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32);(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31);(2,6)(4,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30);(1,3,5,7)(2,4,6,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32);(1,4)(2,7)(3,6)(5,8)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8378975);